lesson 4.3 – triangle inequalities & exterior angles

LESSON 4.3 – TRIANGLE INEQUALITIES & EXTERIOR ANGLES

Homework: 4.3/ 1-10, 12-16

EXTERIOR ANGLE THEOREMAn exterior angle of a triangle…… is equal in measure to the sum of the measures of its two remote interior angles.

remote interior angles Exterior

angle

EXTERIOR ANGLE THEOREM(YOUR NEW BEST FRIEND)

3

2

1 4

exterior angle

remote interioranglesm<1 + m<2 = m<4

m<BCD = m<A + m<B

m<4= m<1+ m<2

EXTERIOR ANGLE THEOREM

EXAMPLES

m<G + 60˚ = 111˚m<G = 51˚

Remote interior angles

Exterior angle

EXAMPLES

x

82°

30° y

Find x & y

x = 68°y = 112°

y = 30 + 82y = 112˚

Using Linear pair:180 = 112 + x68˚ = x

Remote interior angles

EXAMPLESFind m JKM

2x – 5 = x + 70 x – 5 = 70 x = 75

m< JKM = 2(75) - 5m< JKM = 150 - 5m< JKM = 145˚

EXAMPLESSolve for y in the diagram.

Solve on your own before viewing the

Solution

4y + 35 = 56 + y3y + 35 = 563y = 21

y= 7

SOLUTION

EXAMPLESFind the measure of in the diagram shown.1

Solve on your own before viewing the

Solution

40 + 3x = 5x - 1040 = 2x - 1050 = 2x 25 = x

Exterior angle:5x – 10 = 5(25) - 10

m < 1= 65

= 125 – 10 = 115m < 1= 180 -

115

SOLUTION

CHECKPOINT: COMPLETE THE EXERCISES.

SOLUTION

Right Scalene triangle

x + 70 = 3x + 10

70 = 2x + 1060 = 2x30 = x

3 (30) + 10 = 100˚

TRIANGLE INEQUALITIES

Make A Triangle

Construct triangle DEF.

D FF E

D E

D FF E

D E

Make A Triangle

Construct triangle DEF.

D E

Make A Triangle

Construct triangle DEF.

D E

Make A Triangle

Construct triangle DEF.

D E

Make A Triangle

Construct triangle DEF.

D E5 3

13

Q:What’s the problem with this?

A: The shorter segments can’t reach each other to complete the triangle. They don’t add up.

Make A Triangle

Construct triangle DEF.

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

Triangle Inequality Conjecture

Add the two smallest sides; they MUST be larger than the third side

for the triangle to be formed.

TRIANGLE INEQUALITY CONJECTURE

Given any triangle, if a, b, and c are the lengths of the sides, the following is always true:

a + b > ca + c > bb + c > a

The triangle inequality theorem is very useful when one needs to determine if any 3 given

sides will form of a triangle or not.

In other words, if the 3 conditions above are not met, you can immediately conclude that it is not

a triangle.

EXAMPLEThree segments have lengths: a= 3 cm, b= 6 cm, and c = 4 cm.Can a triangle be formed with these measures?

3 + 6 = 9 and 9 > 4

3 + 4 = 7 and 7 > 66 + 4 = 10 and 10 > 3So a triangle can be formed!

EXAMPLEThree segments have lengths: a= 7 cm, b= 16 cm, and c = 8 cm. Can a triangle be formed with these measures?

7 + 16 = 23 and 23 > 8

7 + 8 = 15 , but 15 < 16. This condition is not met because the sum of these two sides is smaller than the third side

16 + 8 = 24 and 24 > 7Since one of the conditions is not met, a triangle cannot be formed.

SIMPLY:

If the two smallest side measures do not add up to be greater than the largest side, then the sides do not make a triangle!

If the two smallest side measures add up to be greater than the

largest side, then the sides make a triangle!

Make A TriangleCan the following lengths form a triangle?1.6 mm5 mm10 mm

2.2 ft9 ft13 ft

5.10 mm3 mm6 mm

8. 8 m7 m1 m

9.9 mm2 mm10 mm

12.1 mm5 mm3 mm

3.5 cm cm4 cm

√𝟐4. 7 ft

15 ft ft

√𝟏𝟑6. 4 ft

7 ft ft

√𝟕

7.10 mm13 mm mm

√𝟓10.

12 mm22 mm mm

√𝟏𝟑

11.5 mm8 mm mm

√𝟏𝟐

In a triangle, the longest side is opposite the largest angle; and the shortest side is opposite the smallest angle.

Side-Angle Conjecture

Side AB is the shortest, because it's across from the smallest angle (40 degrees).  Also, the side BC is

the longest because it is across from the largest angle (80 degrees).

Side-Angle

What’s the biggest side?What’s the biggest angle?What’s the smallest side?What’s the smallest angle?

C

B A

ba

c

bBaA

100°

60°

Side-Angle

92° 42°

46°

ab

c

Rank the sides from greatest to least.bca

Rank the angles from greatest to least.CAB

A

CB

7

5

4

Find x.

Practice

25 + x + 15 = 3x - 10 x + 40 = 3x - 10

40 = 2x - 1050 = 2x25 = x

3x – 10 3(25) – 10 65°

x + 15 25 + 15 40°

Find x and y.

92 = 50 + x40 = x

92 + y = 180y= 88

Exterior angle Linear pair of angles

Find the measures of <‘s 1, 2, 3, & 4

LP: 92 + <1 = 180<1 = 88

LP: 123 + <2 = 180<2 = 57

EA: <4 = <1 + < 2

<4 = 88 + 57<4 = 145

LP: 145 + <3 = 180

<3 = 35

Find the measure of each numbered angle in the figure.

Exterior Angle TheoremSimplify.

SubstitutionSubtract 70 from each side.

linear pairs are supplementary.

Exterior Angle Theorem

Subtract 64 from each side.Substitution

Subtract 78 from each side.

If 2 s form a linear pair, they are supplementary.SubstitutionSimplify.

Subtract 143 from each side.

Angle Sum TheoremSubstitutionSimplify.

Answer:

Find the measure of each numbered angle in the figure.

Answer:

YOUR TURN: